Show that a non-abelain Lie algebra of dimension two has precisely three ideals.

Q: 5. Let F = Q(√2, √3) be the smallest subfield of C that contains √2 and √3. (a) Find a basis for F…

A: Given :  F = ℚ2, 3 be the smallest subfield of ℂ that contains 2 and 3 To find : (a) The basis for F…

Q: Find all maximal ideals of Z_12:

A: Given ring is Z_12

Q: Show that the Galois group of x 4 + x² + 1 is the same as that of x 6-1 and is of order 2.

A: We have to show that the Galois group of is the same as that  of x6 -1 and is of order 2.

Q: 1. Answer the following for the given equation Ax = 0. (a) Find a basis for ColSp(A). (b) Find a…

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Q: Find a basis for span(l - x, x - x2, 1 - x2, 1 - 2x + x2) in P 2

A: We know that Pn is the vector space of all polynomials of degree less than or equal to n. This…

Q: In a commutative ring of characteristic 2, prove that the idempotentsform a subring.

A: Given: Commutative ring of characteristic 2.We need to prove that the idempotents form a subring.

Q: (4)Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to…

A: Given that, K=Z12 and ideals I and J are such that I=([4])=4Z12={0,4,8} J=([6])=6Z12={0,6} Now,…

Q: Find the number of ideals in Cx,y that contains both the polynomials y – (x – 1)³ +1 and p(x) in…

A: We have to find the number of ideals in ℂx,y that contains both the polynomials y-x-13+1 and px in…

Q: _C 16. Suppose R is commutative and let I and J be ideals of R, so R/I and R/J are naturally…

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Q: Let L be a finite dimensional Lie algebra. Any finite dimensional representation of L is completely…

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Q: Show that the Galois group of x" - 1 over any field of characteristic zero is abelian.

A: We have to show that the Galois group of xn -1 over any field of characteristic zero is abelian.

Q: Show that Zmn is ring-isomorphic to Zm ⨁ Zn when m and n are relativelyprime

A: Given: Zmn and Zm⊕Zn. To show:Zmn is ring-homomorphic to Zm⊕Zn when m and n are relatively prime.

Q: Let R be a commutative ring with identity such that every submodule of every free module is free.…

A: Let R be a commutative ring with identity such that every submodule of every free module is free. We…

Q: itely generated projective modules over a local ring are free. jective modules are flat, but the…

A: Sol

Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: 1. Let a = √3+√3 (a) Find ma(z) F raational numbers). the minimal polynomial of a over Q (the field…

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Q: here exists a decomposition field for every polynomial  of positive degree over a field.

A: We have to show that there exists a decomposition field for every polynomial of positive degree over…

Q: is an R-module homomorphism snomorphism if and only if k

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Q: Prove that a finite dimensional Lie algebra over an algebraically closed field is not nilpotent if…

A: Suppose L is a finite-dimensional lie algebra over an algebraically closed field.  To prove that L…

Q: Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual…

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Q: An element x of a ring R is idempotent if x2 = x. (a) Find four idempotent elements of GL(2, R (this…

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Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: Determine the Galois group of x2 +9 over R.

A: The objective of the question is to determine the Galois group of x2+9 over R.

Q: (B) Find all maximal ideals of (Z8, +8,8).

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Q: (10) is maxi

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Q: all possible Jordan Canonical Form of A if the characteristic poly- al is given by (1 – x)*(2 – x)².

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Q: 2. List all possible Jordan Canonical Form of A if the characteristic poly- nomial is given by (1 –…

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Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

Q: Let M be Noetherian left R-modu

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Q: Prove that gl(2, C) is isomorphic to the direct sum of sl(2, C) with C, the 1-dimensional complex…

A: gl(2,C):This is the Lie algebra of all 2×2 complex matrices. It is a 4-dimensional vector space over…

Q: Every submodule of a noetherian (artinian) module is noetherian (artinian).

A: Every submodule of a noetherian (artinian) module is noetherian (artinian).

Q: Compute the Killing form for the Heisenberg algebra and find its radical.

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Q: (10) Let I = (4) be a principal ideal of integer %3| ring Z, Then I is. . ideal * Primary Prime O…

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Q: Show that for any k-algebras A1, A2,..., A,, we have (A1 x A2 x ...x A, )P S A" x A" x as algebras.

A: Let, A1,+,*,A2,+,*,...,Ar,+,* be the K-algebras, where + is the addition from 1 to r o which A1 is…

Q: Theorem: Let A and B be R-dibmodules af left R-modules M and Nrespectively. Then, MXN AXB

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Q: Let R′ be a commutative ring of characteristic 2. Show that                    I ={r∈R′|r^2 =0} is…

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Q: (3) Show that (C SRICs countable or C is antable) is a -algebra on R.

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Q: Let M be an R-module, and let N be an R-submodule  of M. Then M is noetherian (artinian) if and only…

A: Given:M is an R-module, and N is an R-submodule of M. To Prove:M is Noetherian (Artinian) if and…

Q: Let -{[:);}--{:)[:} -{{)}} Both B and C are bases for R2. If {x}s =: -[:], what is {x}c?

A: